Worldly cardinal
In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank Vκ is a model of Zermelo–Fraenkel set theory.[1]
Relationship to inaccessible cardinals
By Zermelo's theorem on inaccessible cardinals, every inaccessible cardinal is worldly. By Shepherdson's theorem, inaccessibility is equivalent to the stronger statement that (Vκ, Vκ+1) is a model of second order Zermelo-Fraenkel set theory.[2] Being worldly and being inaccessible are not equivalent; in fact, the smallest worldly cardinal has countable cofinality and therefore is a singular cardinal.[3]
The following are in strictly increasing order, where ι is the least inaccessible cardinal:
- The least worldly κ.
- The least worldly κ and λ (κ<λ, and same below) with Vκ and Vλ satisfying the same theory.
- The least worldly κ that is a limit of worldly cardinals (equivalently, a limit of κ worldly cardinals).
- The least worldly κ and λ with Vκ ≺Σ2 Vλ (this is higher than even a κ-fold iteration of the above item).
- The least worldly κ and λ with Vκ ≺ Vλ.
- The least worldly κ of cofinality ω1 (corresponds to the extension of the above item to a chain of length ω1).
- The least worldly κ of cofinality ω2 (and so on).
- The least κ>ω with Vκ satisfying replacement for the language augmented with the (Vκ,∈) satisfaction relation.
- The least κ inaccessible in Lκ(Vκ); equivalently, the least κ>ω with Vκ satisfying replacement for formulas in Vκ in the infinitary logic L∞,ω.
- The least κ with a transitive model M⊂Vκ+1 extending Vκ satisfying Morse–Kelley set theory.
- (not a worldly cardinal) The least κ with Vκ having the same Σ2 theory as Vι.
- The least κ with Vκ and Vι having the same theory.
- The least κ with Lκ(Vκ) and Lι(Vι) having the same theory.
- (not a worldly cardinal) The least κ with Vκ and Vι having the same Σ2 theory with real parameters.
- (not a worldly cardinal) The least κ with Vκ ≺Σ2 Vι.
- The least κ with Vκ ≺ Vι.
- The least infinite κ with Vκ and Vι satisfying the same L∞,ω statements that are in Vκ.
- The least κ with a transitive model M⊂Vκ+1 extending Vκ and satisfying the same sentences with parameters in Vκ as Vι+1 does.
- The least inaccessible cardinal ι.
References
- Hamkins, Joel David (2014), "A multiverse perspective on the axiom of constructibility", Infinity and truth, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 25, Hackensack, NJ: World Sci. Publ., pp. 25–45, arXiv:1210.6541, Bibcode:2012arXiv1210.6541H, MR 3205072
- Kanamori, Akihiro (2003), The Higher Infinite, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag
External links
- Worldly cardinal in Cantor's attic
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